Light subgraphs in graphs with average degree at most four

TL;DR

This paper investigates the existence of light subgraphs within graphs that have a small average degree, including specific classes like plane graphs with girth restrictions, expanding understanding of graph substructure properties.

Contribution

It introduces the concept of light graphs in the context of graphs with bounded average degree and explores their presence in plane graphs with girth constraints.

Findings

Identifies conditions under which light subgraphs exist in graphs with average degree at most four.

Establishes bounds on degrees for subgraphs in specific graph classes.

Provides new insights into the structure of plane graphs with girth restrictions.

Abstract

A graph $H$ is said to be {\em light} in a family $\mathfrak{G}$ of graphs if at least one member of $\mathfrak{G}$ contains a copy of $H$ and there exists an integer $\lambda(H, \mathfrak{G})$ such that each member $G$ of $\mathfrak{G}$ with a copy of $H$ also has a copy $K$ of $H$ such that $\deg_{G}(v) \leq \lambda(H, \mathfrak{G})$ for all $v \in V(K)$. In this paper, we study the light graphs in the class of graphs with small average degree, including the plane graphs with some restrictions on girth.